An eulerian walk is a sequence of vertices in a graph such that every edge is...

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

Expert Solution

Colby Messinger answered 1 year ago

Show that a graph without isolated vertices has an Eulerian walk if and only if it...

Show that a graph without isolated vertices has an Eulerian walk if and only if it is connected and all vertices except at most two have even degree.

Prove a connected simple graph G with 16 vertices and 117 edges is not Eulerian.

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G),...

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G), u =v such that there is a trail from u to v which goes over every edge of G. The following sequence of questions is towards a proof of the following: Theorem 1. A connected graph G is semi-Eulerian (but not Eulerian) if and only if it has exactly two vertices of odd degree. Let G be semi-Eulerian with a trail t starting at...

Given a connected graph G with n vertices. We say an edge of G is a...

Given a connected graph G with n vertices. We say an edge of G is a bridge if the graph becomes a disconnected graph after removing the edge. Give an O(m + n) time algorithm that finds all the bridges. (Partial credits will be given for a polynomial time algorithm.) (Hint: Use DFS)

Show that a graph T is a tree if and only if for every two vertices...

Show that a graph T is a tree if and only if for every two vertices x, y ∈ V (T), there exists exactly one path from x to y.

. Determine whether K4 (the complete graph on 4 vertices contains the following: i) A walk...

. Determine whether K4 (the complete graph on 4 vertices contains the following: i) A walk that is not a trail. ii) A trail that is not closed and is not a path. iii) A closed trail that is not a cycle.

Let T be a graph. Suppose there is a unique path between every pair of vertices...

Let T be a graph. Suppose there is a unique path between every pair of vertices in T. Prove that T is a tree. Can I do this using the contraposative? Like Let u,v be in T and since I am assuming T to not be a tree this allows cycles to occur thus the paths must not be unique. Am I on the right track?

Use the fact that every planar graph with fewer than 12 vertices has a vertex of...

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.

(a) Prove the following claim: in every simple graph G on at least two vertices, we...

(a) Prove the following claim: in every simple graph G on at least two vertices, we can always ﬁnd two distinct vertices v,w such that deg(v) = deg(w). (b) Prove the following claim: if G is a simple connected graph in which the degree of every vertex is even, then we can delete any edge from G and it will still be connected.

An Euler Circuit is a circuit that crosses every edge exactly once without repeating. A graph...

An Euler Circuit is a circuit that crosses every edge exactly once without repeating. A graph has an Euler Circuit if and only if (a) the graph is connected and (b) the degree of every vertex is even. Find a lower bound for the time complexity of all algorithms that determine if a graph has an Euler Circuit. In which of the three general categories discussed in Section 9.3 does this problem belong? Justify your answer.

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